How many psychologists have won a Nobel prize?

It depends on how one defines "psychologist." In any case, the answer is "very few," as there is no prize awarded in the field of psychology.

If "psychologist" means a professor of psychology with a degree in psychology, then only one: Daniel Kahneman. His collaborator and co-recipient, Amos Tversky, was not alive to accept the prize, which was awarded in the field of economics.*

Roger Sperry was a psychologist, but held his professorship in psychobiology. That would make two-his being awarded in the field of medicine.

If the definition would include someone with a degree in political science who later became a professor of psychology, then three, counting Herbert Simon, who won his prize in the field of economics.*

Ivan Pavlov, the physiologist whose influential work on classical conditioning has earned him a rank among the top 20th century "psychologists" (as rated by professional psychologists, citations, and references in Introductory Psychology textbooks), also won the prize in the field of physiology. If counted, that would make four.

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*: The answer also depends on how one defines "Nobel prize;" most, but not all, consider the prize in economics a true Nobel. A separate though related organization developed and began awarding the prize 68 years after the first prizes were put in place. Per Wikipedia: "The Prize in Economics is not one of the original Nobel Prizes created by Alfred Nobel's will. … Some critics argue that the prestige of the Prize in Economics derives in part from its association with the Nobel Prizes, an association that has often been a source of controversy."

Even a recipient of the award, Friedrich Hayek, argued against considering the honor as equivalent to the other Nobels: "The Nobel Prize confers on an individual an authority which in economics no man ought to possess. … This does not matter in the natural sciences. Here the influence exercised by an individual is chiefly an influence on his fellow experts; and they will soon cut him down to size if he exceeds his competence. But the influence of the economist that mainly matters is an influence over laymen: politicians, journalists, civil servants and the public generally."